Two subgroups A and B of a group G are almost disjoint if their intersection is trivial — they share only the identity element. Almost disjointness is necessary for independence but not sufficient. Two subgroups can share no elements and still constrain each other.
The category-theoretic definition of independence captures what almost disjointness misses (arXiv:2603.11309). Two subgroups are independent when any pair of endomorphisms — one acting on A, one acting on B — can be extended to an endomorphism of the group they generate. Independence means that transformations of one subgroup can be performed without interfering with the other. The subgroups are truly separate: what you do to one does not constrain what you can do to the other.
Almost disjointness says the subgroups don't share elements. Independence says they don't share structure. A and B can be element-disjoint while their generated group imposes structural relationships between them — relations that force endomorphisms of A to be compatible with certain endomorphisms of B. These structural entanglements are invisible to element-level intersection but visible to the endomorphism extension criterion.
The necessary and sufficient conditions for independence go beyond intersection. They involve how the subgroups sit inside the ambient group — not just what elements they contain but how they combine under the group operation. The position, not just the content, determines independence.